Question

In Problems 13–24, find the domain of each rational function.

$R\left(x\right)=\frac{-2(x^2-4)}{3(x^2+4x+4)}$

Short answer

The domain of the rational function $R\left(x\right)=\frac{-2(x^2-4)}{3(x^2+4x+4)}$is $\{x\overline{)x\in \mathrm{ℝ},x\ne -2\}}$

Step-by-step solution

Step 1. Given

The rational function$R\left(x\right)=\frac{-2(x^2-4)}{3(x^2+4x+4)}$

To find the domain of the given rational function.

Step 2. Factor the denominator.

Factorize the denominator.

$$\begin{gathered} x^2+4x+4=x^2+2x+2x+4 \\ =(x+2)(x+2) \\ ={(x+2)}^2 \end{gathered}$$

Step 3. Use the zero-Product property

By using Zero-Product property,

$x+2=0$

Step 4. Solve the equation

Solve the equations

$$\begin{gathered} x+2=0 \\ x=-2 \end{gathered}$$

The domain of the function is $\{x\overline{)x\in \mathrm{ℝ},x\ne -2\}}$